In recent years, Orthogonal Frequency Division Multiplexing (OFDM) has attracted attention as a candidate for high-data-rate video and multimedia communications. OFDM belongs to a family of transmission schemes called multi-carrier modulation. Multi-carrier modulation is based on dividing a given high bit-rate data stream into several parallel low bit-rate data streams and modulating each stream on separate sub-carriers.
The motivation for using multi-carrier modulation is to overcome the problem of inter-symbol interference (ISI). In wireless channels where radio signals from a transmitter traverse multiple paths to a receiver, all the signal energy does not necessarily arrive at the receiver at the same instant of time. This phenomenon of dispersion in a communication channel causes energy from one symbol duration to spill into succeeding symbol durations.
When a time delay due to dispersion is either a significant fraction of or more than the symbol time duration, the resultant ISI can be detrimental. ISI causes an irreducible error floor that cannot be overcome by simply changing a radio frequency (RF) parameter such as a transmit power, an antenna pattern or a frequency plan.
In an OFDM system, each sub-carrier can be viewed as a flat fading channel. A single tap equalizer can be used to equalize the transmitted signal in case of coherent demodulation. This requires the receiver to have knowledge of the channel on a per-sub-carrier basis.
The discrete baseband time representation of a transmitted OFDM signal is:
                              y                      n            ,            m                          =                              ∑                          k              =              o                                      N              -              1                                ⁢                                    x                              k                ,                m                                      ⁢                          ⅇ                              j                ⁢                                                                  ⁢                2                ⁢                                                                  ⁢                                  π                  ⁡                                      (                                          k                      -                                              N                        2                                                              )                                                  ⁢                                  n                  N                                                                                        (        1        )            where xk,m represents the sub-carrier symbols, N is the number of sub-carriers, k is the modulated symbol index, m is the OFDM symbol index, n is a sub-carrier index, and j represents the square root of −1.
The discrete version of the OFDM baseband signal shown in equation (1) is identical to the Inverse Discrete Fourier Transform (IDFT) of the sub-carrier symbols xn,m. Thus, OFDM modulation is essentially identical to an IDFT operation which may be performed using an Inverse Fast Fourier Transform (IFFT). FIG. 1 (prior art) is a block diagram of a baseband representation of an OFDM transmitter. The OFDM transmitter comprises a serial-to-parallel converter 10, an IFFT processor 12, and a parallel-to-serial converter 14.
FIG. 2 (prior art) is a graph illustrating frequency synchronization of an OFDM signal. The graph illustrates spectral component versus frequency for the case of using four tones.
The sub-carrier data symbols can be estimated at a receiver by taking the Discrete Fourier Transform (DFT) such as a Fast Fourier Transform (FFT) of a received and equalized OFDM symbol using the following equation.
                                          x            ^                                n            ,            m                          =                              1            N                    ⁢                                    ∑                              k                =                o                                            N                -                1                                      ⁢                                          y                                  k                  ,                  m                                            ⁢                              ⅇ                                                      -                    j                                    ⁢                                                                          ⁢                  2                  ⁢                                                                          ⁢                                      π                    ⁡                                          (                                              n                        -                                                  N                          2                                                                    )                                                        ⁢                                      k                    N                                                                                                          (        2        )            
In a time dispersive channel such as one which introduces multipath fading, the signal at the receiver can be written as a convolution of the transmitted signal y and the channel impulse response h. Thus, the received signal rn,m in the discrete time domain can be written as:
                              r                      n            ,            m                          =                                            ∑                              l                =                o                                            L                -                1                                      ⁢                                          h                l                            ⁢                              y                                                      n                    -                    l                                    ,                  m                                            ⁢                                                          ⁢              n                                ≥          L                                    (                  3          ⁢          a                )                                          r                      n            ,            m                          =                                                            ∑                                  l                  =                  o                                n                            ⁢                                                h                  l                                ⁢                                  y                                                            n                      -                      l                                        ,                    m                                                                        +                                          ∑                                  l                  =                                      n                    +                    1                                                                    L                  -                  1                                            ⁢                              A                ⁢                                                                  ⁢                                  h                  l                                ⁢                                  y                                                            N                      -                                              (                                                  l                          -                          n                                                )                                                              ,                                          m                      -                      l                                                                      ⁢                                                                  ⁢                n                                              <          L                                    (                  3          ⁢          b                )            where L is the length of the channel impulse response in units of sample time.
No ISI is observed in received signal when n=L. However when n<L, the signal from the previous OFDM symbol is corrupted by the received signal. If the cyclic prefix is longer than the channel impulse response, then the effect of the previous OFDM symbol is not seen in the usable part of the signal (i.e. the part of the OFDM symbol after the cyclic prefix). From here onward, it is assumed herein that the cyclic prefix is longer than the channel impulse response, thus ignoring the second term in equation (3b). Under this assumption, the demodulated signal in each sub-carrier is given by:
                                                                        s                                  n                  ,                  m                                            =                            ⁢                                                1                  N                                ⁢                                                      ∑                                          k                      =                      0                                                              N                      -                      1                                                        ⁢                                                            r                                              k                        ,                        m                                                              ⁢                                          ⅇ                                                                        -                          j                                                ⁢                                                                                                  ⁢                        2                        ⁢                                                                                                  ⁢                                                  π                          ⁡                                                      (                                                          n                              -                                                              N                                2                                                                                      )                                                                          ⁢                                                  k                          N                                                                                                                                                                                            =                            ⁢                                                1                  N                                ⁢                                                      ∑                                          k                      =                      0                                                              N                      -                      1                                                        ⁢                                                            ∑                                              l                        =                        0                                                                    L                        -                        1                                                              ⁢                                                                  h                        l                                            ⁢                                              y                                                                              k                            -                            l                                                    ,                          m                                                                    ⁢                                              ⅇ                                                                              -                            j                                                    ⁢                                                                                                          ⁢                          2                          ⁢                                                                                                          ⁢                                                      π                            ⁡                                                          (                                                              n                                -                                                                  N                                  2                                                                                            )                                                                                ⁢                                                      k                            N                                                                                                                                                                                                                      =                            ⁢                                                λ                  n                                ⁢                                  x                                      n                    ,                    m                                                                                                          (        4        )            where λn is the DFT of the channel impulse response.
Even though the channel is time dispersive, the effect of the channel can be visualized in the frequency domain (i.e. after the DFT) by a single multiplicative constant λn for each of the sub-carriers. This results because a convolution operation in the time domain translates to a simple multiplication in the frequency domain. A single-tap equalization using a zero-forcing equalizer can be used to estimate the sub-carrier signal using the following equation.
                                          x            ^                                n            ,            m                          =                              s                          n              ,              m                                            λ                          n              ,              m                                                          (        5        )            
However, in order to use equation (5), the receiver needs to know the channel response in the frequency domain (i.e. the values of λn).
The channel response may be estimated either by embedding pilot symbols within data stream or by using preambles. The known pilot or preamble symbols are used to estimate the channel on the given sub-carriers. To estimate the channel in other sub-carriers, a subsequent channel interpolation can be performed.
Existing channel estimation techniques include zero forcing and linear minimum mean square error (LMMSE). Zero-forcing channel estimation can be performed over known pilot and/or preamble symbols by dividing the received symbol by the expected symbol.
                                          λ            ^                    n                =                              s                          n              ,              m                                            x                          n              ,              m                                                          (        6        )            
A disadvantage of a zero forcing estimate is its unreliability in low signal-to-noise ratio (SNR) conditions. For example, a zero-forcing-estimated channel response can be significantly inaccurate if some of the sub-carriers experience a deep fade.
The LMMSE channel estimator is designed to minimize the mean square error between an estimated channel response and an actual channel response. For convenience in formulating the LMMSE estimator, the relationship between received and transmitted symbols in the sub-carriers carrying the pilot/preamble is represented in the following vector form:
                              S          =                                    P              ⁢                                                          ⁢              Λ                        +            W                          ,                              or            ⁢                                                  [                                                                                s                    0                                                                                                                    s                    1                                                                                                ⋮                                                                                                  s                                          M                      -                      1                                                                                            ]                    =                                                    [                                                                                                    p                        0                                                                                    0                                                              …                                                              0                                                                                                  0                                                                                      p                        1                                                                                                                                                                                                0                                                                                                  ⋮                                                                                                                                                                          ⋱                                                              ⋮                                                                                                  0                                                              0                                                              …                                                                                      p                                                  M                          -                          1                                                                                                                    ]                            ⁡                              [                                                                                                    λ                        0                                                                                                                                                λ                        1                                                                                                                        ⋮                                                                                                                          λ                                                  M                          -                          1                                                                                                                    ]                                      +                          [                                                                                          w                      0                                                                                                                                  w                      1                                                                                                            ⋮                                                                                                              w                                              M                        -                        1                                                                                                        ]                                                          (        7        )            where si and pi are the received and transmitted symbols in the sub-carriers carrying the pilot/preamble, and wi represents noise. The noise can be assumed to be additive, Gaussian white noise.
The LMMSE estimate {circumflex over (Λ)} of the channel impulse response is determined by the following equation:{circumflex over (Λ)}=AS=AP{circumflex over (Λ)}+AW  (8)where A is an estimation matrix.
The estimation matrix A is determined by the following equation:A=Rλ[Rλ+(PHP)−1Σ]−1P−1  (9)where Rλ is the covariance matrix of the channel impulse response and Σ is the covariance matrix of the noise vector and usually is a diagonal matrix.
In the absence of noise, the covariance matrix Σ is equal to a zero matrix. If the fading in the different sub-carriers is independent, the covariance matrix Rλ of the channel impulse response is an identity matrix. Under both of these two conditions, the LMMSE estimate is identical to the zero forcing estimate.
A drawback of the LMMSE channel estimation scheme is its requiring knowledge of the correlation between the fades of different sub-carriers to form the covariance matrix Rλ. In most practical systems, this information is not known at the receiver a priori, thus making the LMMSE estimator impractical.
Channel estimation and equalization are fundamental components of wireless communication systems, especially those that have been designed to work in a non-line-of-sight condition. In multi-carrier systems such as OFDM, equalization is relatively straightforward, but sophisticated channel estimation and channel interpolation techniques are presently required.